Pythagorean Theorem Calculator
Pythagorean Theorem
Hypotenuse
c = √(a² + b²)
Leg
a = √(c² - b²)
Common Right Triangles
Related Calculators
The Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This fundamental relationship was known to ancient civilizations and remains one of the most important theorems in mathematics.
The Formula
Solving for Each Side
Hypotenuse (c)
Leg a
Leg b
Pythagorean Triples
A Pythagorean triple consists of three positive integers a, b, and c that satisfy a² + b² = c². These are particularly useful because they give exact integer answers.
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 9 | 40 | 41 | 81 + 1600 = 1681 ✓ |
Special Right Triangles
45-45-90 Triangle
An isosceles right triangle with legs of equal length.
Hypotenuse = leg × √2
30-60-90 Triangle
Half of an equilateral triangle.
Short leg : Long leg : Hypotenuse
Real-World Applications
- Construction: Ensuring corners are square (3-4-5 rule)
- Navigation: Calculating straight-line distances
- Ladders: Finding safe ladder placement angles
- TV Sizes: Diagonal measurement from width and height
- Architecture: Calculating roof pitch and rafter lengths
- Sports: Calculating distance between bases (baseball diamond)
Frequently Asked Questions
What is the 3-4-5 rule in construction?
The 3-4-5 rule uses the Pythagorean theorem to check if corners are square (90 degrees). Measure 3 feet on one side, 4 feet on the other. If the diagonal is exactly 5 feet, the corner is square. Multiples work too: 6-8-10, 9-12-15, etc.
How do you find the hypotenuse?
Square both legs, add them together, then take the square root. If legs are 3 and 4: 3 squared + 4 squared = 9 + 16 = 25. Square root of 25 = 5. The hypotenuse is always the longest side, opposite the right angle.
Can the Pythagorean theorem be used for non-right triangles?
No, the standard formula a squared + b squared = c squared only works for right triangles. For other triangles, use the Law of Cosines: c squared = a squared + b squared - 2ab times cos(C), where C is the angle opposite side c.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy a squared + b squared = c squared. Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. They're useful because they give exact integer answers without square roots or decimals.